nLab Sandbox

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$$\begin{array}{l} \big(\widehat{\rho}'(g^{-1})\widehat{\rho}(g)\big) \rho(h) \big(\widehat{\rho}'(g^{-1})\widehat{\rho}(g)\big)^{-1} \\ \;=\; \widehat{\rho}'(g^{-1})\widehat{\rho}(g) \rho(h) \widehat{\rho}(g^{-1})\widehat{\rho}'(g) \\ \;=\; \widehat{\rho}'(g^{-1}) \rho(g h g^{-1}) \widehat{\rho}'(g) \\ \;=\; \end{array}$$

Particle-hole symmetry.

Original

• S. M. Girvin: Particle-hole symmetry in the anomalous quantum Hall effect, Phys. Rev. B 29 (1984) 6012(R) [doi:10.1103/PhysRevB.29.6012]

$\neq \gt 1$ may be understood as $\nu = 1$ for majority spin polarization with $\nu - 1$ for the minority polarization (Haldane 1983 doi:10.1103/PhysRevLett.51.605)

Here

• “$\bullet$” means that the corresponding combination of $(k,q)$ is not admissible, in that $k q$ is odd or the Gauss sum $\sum_{n=0}^{k-1} e^{ \tfrac{\pi \mathrm{i}}{k} q n^2 } = 0$.

• “$\circ$” means that $\tfrac{q}{k} \gt 2$

So:

1. in the column of $q = 1$ all odd $k$ are excluded and all even $k$ are admissible,

2. in the column of $q = 2$ all odd $k$ are excluded and exactly every second even $k$ is admissible,

3. in rows of odd $k$ the odd $q$ are excluded, and for even $q = 2 r$ the result is admissible iff the Jacobi symbol $(r \vert k) \neq 0$.

   Since the Jacobi symbol $(r \vert k)$ vanishes iff $gcd(r,k) \neq 1$, this means that, at least for odd $k$, exactly only the reduced fractions appear.